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A new extension of the exponential distribution, proposed by Nadarajah and Haghighi (Statistics 45, 543–558 (2011)), is an alternative to the gamma, Weibull and generalized-exponential models, it is also known as NH distribution. The maximum likelihood and Bayes inferential approaches for estimating the unknown two-parameters and some lifetime parameters such as survival and hazard rate functions of the NH distribution in presence of progressive first-failure censored sampling are considered. Based on observed Fisher’s information matrix, the approximate confidence intervals for the two-parameters, and any function of them, are constructed. Using Lindley’s approximation and Markov chain Monte Carlo methods under the assumption of conjugate gamma priors, the Bayes estimates and associate highest posterior density credible intervals for the unknown parameters and reliability characteristics are developed based on squared error loss function. Although the proposed estimators cannot be expressed in explicit forms, these can be easily obtained through the use of appropriate numerical techniques. A Monte Carlo simulation study is carried out to examine the performance of proposed methods. Using different optimality criteria, an optimal censoring scheme has been suggested. Finally, a real data set is analyzed for illustration purposes.

Generalized progressive hybrid censoring plan proposed to overcome the limitation of the progressive hybrid censoring scheme is that it cannot be applied when very few failures may occur before pre-specified terminal time 𝑇. In this paper, the estimating problems of the model parameters, reliability and hazard rate functions of Nadarajah-Haghighi distribution when a sample is available from generalized progressive hybrid censoring have been considered. The maximum likelihood and Bayes estimators have been obtained for any function of the model parameters. Approximate confidence intervals for the unknown parameters and any function of them are constructed. Using independent gamma informative priors, the Bayes estimators of the unknown parameters are derived under the squared-error loss function. Two approximation techniques, namely: Lindley approximation method and Metropolis Hastings algorithm have been used to carry out the Bayes estimates and also to construct the associate highest posterior density credible intervals. The performance of the proposed methods are evaluated through a Monte Carlo simulation study. To select the optimum censoring scheme among different competing censoring plans, different optimality criteria have been considered. A real-life dataset, representing the failure times of electronic devices, is analyzed to demonstrate how the applicability of the proposed methodologies in real phenomenon.

One of the most important asymmetrical probability distributions that recently presented as an extension of the conventional exponential distribution is the alpha power exponential distribution. It may be compared to various asymmetrical well-known models, such as Weibull and gamma distributions. As a result, using an adaptive progressive Type-II hybrid censoring scheme, this paper investigates the estimation problems of the alpha power exponential distribution. Maximum likelihood and Bayesian methods are used to estimate unknown parameters, reliability, and hazard rate functions. Under the assumption of independent gamma priors and symmetric loss function, Bayesian estimators are examined. The Bayesian credible intervals and estimated confidence intervals of the relevant values are also calculated. The various estimating approaches are evaluated using a simulation study that considers various sample sizes and censoring schemes. Furthermore, numerous optimality criteria are examined, and the best progressive censoring schemes are offered. Finally, for an explanation, two real data sets from engineering and chemical fields are provided to show the applicability of the asymmetrical alpha power exponential distribution. The Bayesian method for estimating the parameters and reliability indices of the alpha power exponential distribution is recommended based on numerical results, especially when the number of observed data is small.

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